ANNALES DE L I. H. P., SECTION C

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1 ANNALES DE L I. H. P., SECTION C JEAN-PIERRE AUBIN ROGER J. WETS Stable approximations of set-valued maps Annales de l I. H. P., section C, tome 5, n o 6 (1988), p < 5_6_519_0> Gauthier-Villars, 1988, tous droits réservés. L accès aux archives de la revue «Annales de l I. H. P., section C» ( implique l accord avec les conditions générales d utilisation ( Toute utilisation commerciale ou impression systématique est constitutive d une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

2 Ann. Inst. Henri Poincaré, Vol. 5, n 6, 1988, p Analyse non linéaire Stable approximations of set-valued maps Jean-Pierre AUBIN and Roger J. B. WETS CEREMADE, Universite de Paris-Dauphine and University of California at Davis, U.S.A. ABSTRACT. - The celebrated Lax principle "stability and consistency implies convergence" is adapted to the case of nonlinear equation and even inclusions (multivalued equations) through a convenient. concept of stability. It requires the definition of "contingent derivative" of single or set-valued maps and states that a family of maps is stable if and only if the inverses of their contingent derivatives are bounded. An extension of the Banach-Steinhauss theorem to the set-valued analogues of continuous linear operators is also provided, as well as relations between pointwise and graph convergence of sequences of setvalued maps. Key words : Stability, set-valued map, contingent derivative, graph convergence. RESUME. - Le celebre principe de Lax : «stabilite et consistence impliquent convergence» est adapte au cas des equations non lineaires et meme des inclusions (equations multivoques) grace a une definition adequate de la stabilite. Celle-ci est basee sur le concept de «derivee contingente» d une correspondance. On dit qu une famille de correspondances est stable si les inverses de leurs derivees contingentes sont bornes. (*) This research was conducted when the authors were visiting IIASA, International Institute for Applied Systems Analysis. Annales de l lnstitut Henri Poincaré - Analyse non linéaire Vol. 5/88/06/519/17/$ 3,70/ Gauthier-Villars

3 520 J.-P. AUBIN AND R. J. B. WETS On demontre aussi une adaptation du theoreme de Banach-Steinhauss au cas des processus convexes fermes, qui sont les analogues multivoques des opérateurs lineaires continus, et on examine les relations entre convergence simple et convergence graphique d une suite de correspondances. INTRODUCTION Let us consider two Banach spaces X and Y and a set-valued map F from X to Y. An element yo e Y being given, we consider a solution xo E X to the inclusion We shall approximate such a solution xo by solutions x" inclusions to the where Xn and Yn are Banach spaces, Fn are set-valued maps from Xn to Y nand Y n are given. We extend Lax s celebrated assertion that "consistency and stability imply convergence" [see e. g., Aubin ( 1972)] still holds true for solving very general inclusions, since we assume only that the graphs of the setvalued maps F and Fn are closed. Namely, we prove that if Xm Yn are approximations of X and Y, if y" "approximates" yo and if the Fn are "consistent with F", an adequate "stability property" of the set-valued maps F" implies the convergence of some solutions xn to (~)n to xo. We shall also derive an estimate of the error between xn and xo, which is of the same order as the error between yo and yn and the consistency error between F and Fn. In the process, we obtain an adaptation of the Banach-Steinhauss Theorem to closed convex processes, the set-valued analogues of continuous linear operators. The tool used to define the "stability" of the set-valued maps Fn is the "contingent derivative" introduced in Aubin (1981) [see Aubin-Ekeland ( 1984), Chapter 7]. Stability of the Fn s means, roughly speaking, that the norms of the inverses of the contingent derivatives of the Fn are uniformly bounded. The techniques used in the proof are the ones used for proving inverse function theorems for set-valued maps [Aubin (1982), (1984), Annales de l lnstitut Henri Poincaré - Analyse non linéaire

4 STABLE APPROXIMATIONS 521 Aubin-Ekeland (1984), Aubin-Frankowska (1987), Frankowska (1986)]. They are based on Ekeland s Theorem. 1. STABILITY AND CONSISTENCY IMPLY CONVERGENCE Let X be a Banach space. We consider a family of Banach spaces Xn and operators X) which are right invertible. We denote by XJ a right-inverse of pn. The family (Xn, Pm is a convergent approximation of X if If a Banach space Xo is contained in X with a stronger topology, we denote by the "error function". The Banach spaces Xn are supplied with the norm We then consider convergent approximations (Xn, pn, r n) and (Yn, qn, sj of the Banach spaces X and Y. We also consider set-valued maps F from X to Y and Fn from X" to Yn. We denote yo; F n) the lack of consistency of F~ at (xo, yo), defined by: We say that Fn are consistent with F at (xo, yo) if ~ (xo, yo, Fn) ~ 0. As announced in the introduction, the definition of "stability" we suggest involves the concept of "contingent derivative". Let us begin by defining the concept of contingent cone to K at x E K, introduced by G. Bouligand in the 30 s. We say that v E X belongs to the "contingent cone" TK (x) to K at x if and only if It is a closed cone (not necessarily convex), equal to X whenever x belongs to the interior of K, which coincides with the tangent space when K is a smooth manifold and with the tangent cone of convex analysis when K is a convex subset. [See Aubin-Ekeland ( 1984), Chapter 7, for more details.] When F is a set-valued map from X to Y, the "contingent derivative" DF (xo, yo) at a point (xo, yo) of the graph of F is the set-valued map Vol. 5, n

5 - Let 522 J.-P. AUBIN AND R. J. B. WETS from X to Y defined by v belongs to DF (xo, yo) (u) if and only if (u, v) belongs to the contingent cone to the graph of F at (xo, yo). ( 1. 4) In other words set-valued maps whose graph are cones are positively homogeneous: they are actually called "processes" [see Rockafellar ( 1967), ( 1970)]. Hence contingent derivatives are "closed processes". One can also prove that v belongs to DF (xo, Yo)(u) if and only if We are now ready to define "stable families" of set-valued maps Fn. DEFINITION that the approximations (xo, yo) belong to the graph of F and suppose pn, rn) and (Y", q", s") of X and Y are given. We say that a family of set-valued maps F" : Y" is stable around (xo, yo) if there exist constants c > 0, r~ > 0 and a E ]0, 1[ such that, for all (xn, yn) E Graph Fn satisfying for all vn E Yn, there exist un E X" and wn E Yn satisfying STABILITY. THEOREM Let X and Y be Banach spaces and p", rn), (Y", qn, sn) two.f amilies of convergent approximations. Let us consider set-valued maps F from X to Y and F" from X" to Yn with closed graphs. Let xo be a solution to the inclusion Suppose the set-valued maps Fn are consistent with F and stable around (xo, If qnyn converges to yo, then there exist solutions xn to the inclusions such that p" x" converges to xo. Furthermore, threre exists a constant l > 0 such that, for all yn, n and x" E F-1n (n) satisfying yo, qnyn ~ yo and pn xn ~ xo, we have Annales de l lnstitut Henri Poincaré - Analyse non linéaire

6 STABLE APPROXIMATIONS 523 In particular, we deduce that Remark. - Stability is necessary. When the vector spaces Xn are finite dimensional, condition ( 1. 8) is actually equivalent to the stability of the F n. Indeed, let vn E Y n be fixed and set yn : for all h > o. By ( 1. 8), there exists xh~f-1n (yn+hv) such that Hence uh : is bounded by l ( 1 + E) ~ Since the dimension of Xn is finite, a subsequence (again denoted) M~, converges to some u, a solution to Hence the Fn s are stable. D Remark. - By taking y" = sn yo and Fn : = sn F pn, we obtain the estimates The right-hand side converges to 0 when F is lower semicontinuous. Remark. - First stability criteria. The set-valued maps F~ are stable when, for instance, their contingent derivatives DF~ (xn, y") are surjective and when the norms of their inverse are uniformly bounded. The norm of DFn (xn, is defined by D The question arises whether an extension of the Banach-Steinhauss Theorem could provide stability criteria. For that purpose we need to introduce the set-valued analogues of continuous operators, which are the set-valued maps whose graphs are closed convex cones (instead of closed vector. spaces). They are called "closed convex processes". A map A with closed graph is a closed convex process if and only if Contingent derivatives are not always closed convex processes. When the spaces are finite dimensional, the lower semicontinuity of Vol. 5, n

7 524 J.-P. AUBIN AND R. J. B. WETS (x, y) -Graph DF (x, y) at (xo, yo) implies that DF (xo, yo) is a closed convex process [see Aubin-Clarke ( 1977)]. When the contingent derivative is not a closed convex process, we can consider closed convex processes contained in it. For instance, we could work with the asymptotic derivative, introduced by Frankowska ( 1983), (1985). If A is a closed process from X to Y, the set-valued map A ~ from X to Y is defined by Since the graph of A ~ is a Minkowski difference (or the asymptotic cone of Graph A), it is a closed convex cone. Hence A~ is a closed convex process contained in A. Consequently, the "asymptotic contingent derivative" D~ F (x, y) defined by is a closed convex process contained in the contingent derivative. It also contains always the derivative CF (x, y), whose graph is the Clarke tangent cone to the graph of F at (x, y), introduced in Aubin (1982) [see also Aubin-Ekeland, ( 1984), Chapter 7]. In any case, let us consider a family of closed convex processes Ari from Xn to Y~ such that UNIFORM BOUNDEDNESS THEOREM Let us assume that closed convex processes An are surjective and satisfy, for all An n ( (xo, B) ~ V v E Y, there exists u" E A" 1 (s" v) such that sup~pn u" I + ~. ( 1. 15) Then the family of set-valued maps Fn is stable. Proof. - We consider the functions pn and p defined by n the and Since An is a convex process and s" is linear, we deduce that pn is convex and positively homogeneous (sublinear). Since each set-valued map is a closed convex process whose domain is the whole space, the function pn is continuous, thanks to the Robinson-Ursescu s [Robinson ( 1976), Ursescu ( 1975)] theorem, an extension of the Banach Closed Graph Annales de I Institut Henri Poincaré - Analyse non linéaire

8 Let STABLE APPROXIMATIONS 525 Theorem. Then the function p is lower semicontinuous, convex and positively homogeneous. Assumption ( 1. 15) implies that it is finite. We thus deduce from Baire s Theorem that it is continuous, and thus, that there exists a constant c > 0 such that i. e. that for all v E Y, there exists such that By taking we deduce that the family of F n s is stable. D Remark. - If the Banach spaces Xn are reflexive, we do not need the Robinson-Ursescu Theorem, since it is easy to check that the function p" is lower semicontinuous, and thus, continuous. D We also mention another useful consequence of the Uniform Boundedness Theorem. - THEOREM us consider a metric space U. Banach spaces X and Y, and a set valued-map associating to each u~u a closed convex process u H A (u) : X -~ Y. Let us assume that the family of convex processes ~ A (u), u E U ~ is bounded, in the sense that Then the following are equivalent (i) the set-valued map u H Graph A (u) is lower semicontinuous ; (ii) the set-valued map (u, x) H A (u) (x) is lower semicontinuous. Proof. - Condition (ii) implies condition (i), even when the family { A (u) ~ is not bounded. For proving the converse, consider a sequence of elements (un xn) converging to (x, u) and choose an arbitrary element y in A (u) (x). We have to approximate it by elements Since u H Graph A (u) is lower semicontinuous, we can approximate (x, y) by elements Y n) E Graph A (un). By Theorem 1. 2, applied to the there exists a constant l > 0 such that Hence we can choose zn E A (x" - xn) such that ) ) (1 + E). Therefore yn : = y" + zn does belong to A (un) (xj and converges to y because z~ converges to 0 and yn to y. D Remark. - Dual stability criteria. Closed convex processes, as continuous linear operators, can be transposed. Let A be a set-valued map from X to Y. Its transpose A* from Y* to X* is the closed convex process defined by pea * (q) if and only if Vol. 5, n

9 526 J.-P. AUBIN AND R. J. B. WETS In other words, p belongs to A * (q) if and only if (p, - q) belongs to the polar cone of Graph A. [See Rockafellar ( 1967), (1970), Aubin-Ekeland ( 1984). ] Many properties of transposition of continuous linear operators can be extended to closed convex processes. For instance, q belongs to (Im A) - if and only if Therefore, if the vector space Y is finite dimensional, A is surjective if and only if the kernel A * -1 (0) of its transpose is reduced to 0. We also check that in this case where B* is the unit ball of Y*. It is easy to deduce from Theorem 1. 2 the following COROLLARY 1. l. - Let us consider closed convex processes An contained in yn) for all y") in Graph Fn n ((xo, B). Let us assume that their transpose An satisfy Then the family of Fn s is stable. Remark. - Graph and pointwise convergence of set-valued maps. We consider now the case when X" = X and Yn = Y for all n. Let F" be a family of set-valued maps from X to Y. We can define the convergence of the set-valued maps Fn either from the convergence of their graphs (graph convergence) or from the convergence of their values F n (xn) (pointwise convergence). We recall the following definitions of the Kuratowski upper and lower limits of a sequence of subsets Kn of a Banach space K". We denote by F# the set-valued map defined by and by Fb the set-valued map defined by Annales de l lnstitut Henri Poincaré - Analyse non linéaire

10 STABLE APPROXIMATIONS 527 The following relations follow directly from the definitions It is also easy to check that The Stability Theorem (applied to the maps F; 1 instead of the maps F n) implies the equality of F and F#. PROPOSITION 1. l., - Let us assume that the set-valued maps stable around (xo, yo) E Graph Fb. Then yo belongs to lim inf Fn (x"). xn - xo This point of view, that leads to replacing pointwise by graph convergence was already found to be advantageous in the "epigraphical" setting, i. e., for the set-valued functions x H f (x) + R + where f is an extended real valued function defined on the space X. The results reported in the literature are mostly of topological nature, cf. Salinetti and Wets ( 1976), Dolecki, Salinetti and Wets (1983); for more about epi-convergence and graph convergence consult Attouch (1984). In a subsequent paper, we develop the applications of these results to epigraphical maps, and show how they can be used to obtain approximation and stability results of a quatitative nature for variational problems. are p 2. THE LINEAR CASE WITH CONSTRAINTS We shall deduce the stability theorem 1. 1 from a simpler statement. We consider two Banach spaces Z and Y, a continuous linear Y) and a subset K of Z. We consider the problem (a linear equation with constraints) find xo E K a solution to A x =yo. Remark. - By taking Z : = X x Y, K : = Graph F, A : _ ~cy, the projection from X x Y to Y, we observe that inclusion (*) is a particular case of this problem. D Vol. 5, n

11 We Supplied 528 J.-P. AUBIN AND R. J. B. WETS We approximate this problem by introducing (i) convergent approximations pn, rn) and (Yn, qn, sn) spaces Z and Y; (ii) subsets Kn c zn, (iii) continuous linear opertors An 6 Jzf (Zn, YJ. We use the following approximate problems: of the The "convergence" of Yn to yo, of Kn to K at xo and of An measured by the following to A is A") are - DEFINITION shall say that these approximations "stable" if and only if there exist constants c > 0, r~ > 0 and ae ]0,1 such that for all n, for all xn E K" satisfying r~ and for all v" E Y", there exist un E Z and wn E Yn satisfying THEOREM Let us assume that the subsets Kn are closed. Assume that the approximations are stable. Then, if ( y0- qn yn d (xo, pn Kn) and c (A, An) converge to 0, there exist solutions xn~kn to An xn = yn which converge to xo. Furthermore, there exists a constant I such that, for all xn E K" such that p" xn converges to xo, we have In particular, Proof of theorem with the metric xn) : K" is complete. We apply Ekeland s theorem to the continuous function V" defined on Kn by Let E ( 1- a)/c be chosen. We take x" E K" such that () converges to 0. Annales de l Institut Henri Poincaré - Analyse non linéaire

12 STABLE APPROXIMATIONS 529 Therefore, Ekeland s theorem implies the existence of xn E K" satisfying The first inequality implies that The error converges to 0 since and Consequently, for n large enough, the Pn xn belong to B (xo, r~). By the stability assumption, associate with vn : = yn-anxn elements and we can By the very definition of the contingent cone, we assign to any h > 0 (which will converge to 0) elements where 0 (h) converges to 0 with h. By taking such an x", from the second inequality of (2. 6), we obtain This implies that Vol. 5, n

13 530 J.-P. AUBIN AND R. J. B. WETS By letting h converge to 0, we obtain Since (X+EC 1, this implies that Therefore, and Since this inequality is true for ( 1- a) /c, so that we can let E converge to By taking x" E Kn such that ) ) d (xo, p" Kn) ( 1 + j3) and letting 03B2 converge to 0, we obtain the estimate (2. 5). D Proof of the stability theorem 1. l. - We take Z : = X x Y, K = Graph = Graph Fn and An : =03C0Yn. We observe that c (A, since, for all Yn)~ The stability of the set-valued maps Fn is just the same as the stability of their graphs with respect to the projections ~cy and 1ty n. If (xm Yn) is in the graph of F m we deduce that Finally, we can estimate the distance between (xo, yo) and the image of the graph of Fn by pn x qn in the following way. Indeed, (pn x qn) Graph Fn Graph (qn = qn is ~7n On ~7n one has qn pn qn Fn xw Hence Theorem 1. 1 follows from Theorem D where the domain of Annales de l lnstitut Henri Poincaré - Analyse non linéaire

14 Assume STABLE APPROXIMATIONS A STABILITY CRITERION We devote this section to criteria implying that a family of subsets Kn is stable. For simplicity, we consider the case when Zn : =Z, Y~=Y, qn = id, Pn = id and An = A. It is time to recall that the Kuratowski lim inf is the set of x s such that x = lim xn where xn E Kn. The stability assumption (2. 3) implies implicity that xo belongs lim inf of the subsets Kn. We consider now the lim inf of the contingent cones to the and we address the following question: under which conditions does the "pointwise surjectivity assumption" imply the stability of the K". The next result answers this question when the dimension of Y is finite, unfortunately. - PROPOSITION that T (xo) is convex and that AT (xo) Y. = Let us assume that there exists a space Y such that the injection from Y to H is compact. There exists a constant c > 0 such that, for all a E ]o,1 [, there exist r~ > 0 and N >_ 1 with the following property: there exist solutions ~c" E TKn (x") and W" E Y to Remark. - When Y is finite dimensional, we can take H = Y. Proof. - Let S denote the unit sphere of Y, which is relatively compact in H. Hence there are p elements vi such that the balls vi + (a/2) BH cover S. Since T(xo) is convex and AT (xo) = Y, Robinson-Ursescu s Theorem implies the existence of a constant ~, > 0 such that we can associate with any vi E S an ui E T (xo) satisfying I ~,. By the very definition of T (xo), we can associate with a E ]o,1 integers N1 and ~i > 0 such that V x" E Kn n (xo + B), there exist un E TKn (xn) satisfying D Vol. 5, n

15 532 J.-P. AUBIN AND R. J. B. WETS Let N : = max Ni and r~ : = min We take n >_ N and xn E Kn (~ (Xo+l1 B). Let v belong to Y. There exists such that (where c : =~,+1/2IIA H)) and that This proves our claim. D This result justifies a further study of the lim inf of contingent cones to TKn (xn). We introduce the cone CKn (xo) of elements v such that When all the Kn s are equal to K, then lim inf Kn = K and CKn (xo) coincides with the Clarke tangent cone to K at xo. It is clearly a closed convex cone: indeed, let vi, and v2 belong to CKn (xo), xn E Kn a sequence converging to xo and ~ hn 0+. There exists a sequence v 1 n converging to v 1 such that xn : belongs to Kn for all n. Since xn also converges to xo, there exists a sequence v2n converging to v2 such that for all n. Hence xn + hn + v2n) xn = + hn v2n E Kn for all n and v1n + U2n converges to vl + v2. Then belong to CKn (xo). A slight modification of a result of Aubin-Clarke (1977) implies the following relations between T (xo) and CKn (xo). PROPOSITION Assume that Z is reflexive and that the subsets Kn are weakly closed. Then Annales de l Institut Henri Poincaré - Analyse non linéaire

16 STABLE APPROXIMATIONS 533 Proof. - Let v belong to lim inf TKn (xn). Then, for any E > o, there exists N such that Let us set gn (t) : =d(xn+tv, Kn). By Proposition , p. 178 of Aubin- Cellina ( 1984), where yn E K" is a best approximation of x" + tv. Let xo" E K denote a best approximation of xo. Since when x" E (xo + (rt/2) B) n K" and t _ r~/4 ( I v II, we deduce that the function which is almost everywhere differentiable, satisfies By integrating from 0 to h, we deduce that for all n? N and D The converse is true when the dimension of Z is finite or when the subsets K" are convex. More generally, we introduce the following "weak contingent cones" T (x) defined in the following way: v belongs to TK (x) if and only if there exist a sequence hn 0 + and a sequence w" converging weakly to v such that belongs to K for all n. We see at once that and that they coincide when the dimension of Z is finite or when K is convex: indeed, in this case, TK (x) and T (x) are the closure and the weak closure of the convex cone spanned by K - x, which thus are equal. We then obtain the following trivial inclusion: PROPOSITION Assume that Z is reflexive, then Vol. 5, n

17 - _ 534 J.-P. AUBIN AND R. J. B. WETS Proof. - Assume that v belongs to CKn (xo). Then, for all E > o, there exist ~ > 0, N and 03B2>0 such that, for all n >_ N and xn~kn (1 (xo + ~ B), Let us fix such an n _>_ N and Let such that _ 2 E and since and set v~ : the space is reflexive, a subsequence of vn converges weakly to some Such a vn belongs to TKn (xn). Hence d (v, to 0. 0 (xn)) converges REFERENCES H. ATTOUCH, Variational Convergence for Functions and Operators, Applicable Mathematics Series Pitman, London, J.-P. AUBIN, Approximation of Elliptic Boundary-Value Problems, Wiley-Intersecience, J.-P. AUBIN, Contingent Derivatives of Set-Valued Maps and Existence of Solutions to Nonlinear Inclusions and Differential Inclusions, Advances in Mathematics, Supplementary studies, L. NACHBIN Ed., 1981, pp J.-P. AUBIN, Comportement lipschitzien des solutions de problèmes de minimisation convexes, C.R. Acad. Sci. Paris, T. 295, Series I, 1982, pp J.-P. AUBIN, Comportement lipschitzien des solutions de problèmes de minimisation convexes. In Non linear Parial Differential Equations and their Applications, Collège de France seminar IV, 81/82, 1983, pp. 1-18, Research Notes in Mathematics, Pitman, London. J.-P. AUBIN, Lipschitz Behavior of Solutions to Convex Minimization Problems, Mathematics of Operations Research, Vol. 8, 1984, pp J.-P. AUBIN and A. CELLINA, Differential Inclusions, Springer-Verlag; Grundlehren der Mathematische Wissenschaften, Vol. 264, 1984, pp J.-P. AUBIN and F. H. CLARKE, Monotone Invariant Solutions to Differential Inclusions, J. London Mathematical Society, Vol. 16, 1977, pp J.-P. AUBIN and I. EKELAND, Applied Nonlinear Analysis, Wiley-Interscience, J.-P. AUBIN and H. FRANKOWSKA, On inverse Function Theorems for Set-Valued Maps, J. Mathématiques pures et appliquées T. 66, 1987, pp F. H. CLARKE, Optimization and Nonsmooth Analysis, Wiley-Interscience, Sz. DOLECKI, G. SALINETTI and R. WETS, Convergence of Functions: Equi-Semicontinuity, Transactions of the American Mathematical Society, Vol. 276, pp I. EKELAND, On the Variational Principle, J. Mathematical Analysis and Applications, Vol. 47, 1974, pp H. FRANKOWSKA, 1974, Inclusions adjointes associées aux trajectoires minimales d une inclusion différentielle, C.R. Acad. Sci. Paris, T. 206, Series I, H. FRANKOWSKA, A Viability Approach to the Skorohod Problem, Stochastics, Vol. 14, 1985, pp H. FRANKOWSKA, An Open Mapping Principle for Set-Valued Maps, Journal M. A. Appl., Vol. 127, 1987, pp S. ROBINSON, Regularity and Stability for Convex Multivalued Functions, Mathematics of Operations Research, Vol. 1, 1976, pp R. T. ROCKAFELLAR, Monotone Processes of Convex and Concave Type, Memoirs of American Mathematical Society, No. 77, R. T. ROCKAFELLAR, Convex Analysis, Princeton University Press, Annales de l Institut Henri Poincaré - Analyse non linéaire

18 STABLE APPROXIMATIONS 535 R. T. ROCKAFELLAR, La théorie des sous-gradients, Presses de l Université de Montréal, Montréal, G. SALINETTI and R. WETS, On the Relation Between Two Types of Convergence for Convex Functions, J. Mathematical Analysis and Applications, Vol. 60, 1977, pp C. URSESCU, Multifunctions with Closed Convex Graph, Czechoslovakia Mathematics J., Vol. 25, 1975, ( Manuscrit reçu le 12 aoft 1987.) Vol. 5, n